On the almost unperturbed Schrödinger pair of operators
A pair (U,V) of (unbounded) self-adjoint operators on a Hilbert space is called an almost unperturbed Schrödinger pair with parameter \(a\in {\mathbb{R}}\setminus \{0\}\) if there is a trace class operator D such that \[ i[U,V]\subset aI+D \] (where the domain of the commutator [U,V] is assumed to be large enough). This notion is related to the classical Schrödinger pair (satisfying the Heisenberg commutation relation, i.e. \(D=0)\) and to the case of almost commuting operators (i.e., \(a=0)\). The theory of pairs with these properties is well established, and in this paper the author starts to develop the theory of almost unperturbed Schrödinger pairs. It is shown, e.g., that under certain conditions an almost unperturbed Schrödinger pair has a singular integro-differential model, i.e., U is multiplication by the independent variable on \(L_ 2({\mathbb{R}})\) whereas V is differentiation plus a certain perturbation. Moreover, results on symbols, the principal distribution, a trace formula, and the determining function are established. In the last section perturbations of n-parameter groups of unitary operators are investigated.
perturbations of n-parameter groups of unitary operators
scientific article; zbMATH DE number 4111387